For two polynomials \(F(x,y),\, G(x,y)\in \mathbb {F}_q[x,y]\)  of positive degree in \(y\) , the resultant \(Res(F,G,y)\)  of \(F\)  and \(G\)  with respect to \(y\)  is the resultant of \(F\)  and \(G\)  when considered as polynomials in the single variable \(y\) . In this case, \(Res(F,G,y)\in \mathbb {F}_q[x] \cap \langle F,G\rangle \) , where \(\langle F,G\rangle \)  is the ideal generated by \(F\)  and \(G\) . Thus any pair \((a,b)\)  with \(F(a,b)=G(a,b)=0\)  is such that \(Res(F,G,y)(a)=0\) . For more information on resultants and elimination theory, the reader can refer to [1]}.
